Induced representations of locally profinite groups

Abstract

Let G be a separable, locally prolinite (i.e., locally compact and totally disconnected) group with centre Z, and L an open subgroup of G containing Z and such that L/Z is compact. This note contains a couple of results on the structure of (smooth) representations of G induced from irreducible (therefore finite-dimensional) representations of L. The first demonstrates the equivalence of various “finiteness” conditions on the induced representation, while the second gives a finiteness property in the general case. An immediate corollary of the second is the fact (well known for p-adic reductive groups) that a given irreducible representation of L can occur in at most finitely many irreducible supercuspidal representations of G. In the case, for example, where G is a general linear group over a nonArchimedean local field and L is a maximal compact mod centre subgroup, it has long been appreciated that an irreducible representation of G induced from L is automatically supercuspidal. Indeed, this observation is the starting point of the current theory of classification of supercuspidals, see, for example, [6]. Our results answer the frequently encountered question as to what happens when the induced representation is not irreducible. As the extreme generality of our hypotheses indicates, the results here are straightforward, even formal, in nature. It is this essential simplicity, as much as the results themselves, which prompts me to write this rather easy little note.